# Standing waves

A stationary or standing wave is one in which the amplitude varies
from place to place along the wave. Figure 1 is a diagram of a stationary wave. Note that
there are places where the amplitude is zero and, halfway between, places where the
amplitude is a maximum; these are known as **nodes**
(labelled N) and **antinodes** (labelled A) respectively.

(See
Figure 1)

A **node** is a place of zero amplitude

An **antinode** is a place of maximum amplitude
The distance between successive
nodes, and successive antinodes, is half a wavelength. (l/2)

The amplitude of the
points on a stationary wave varies along the wave. In Figure 1 the amplitude at point 1 is
a_{1}, that at point 2 is a_{2} and that at point 3 is a3. The displacement (y) at
these points varies with time.

Any stationary wave can be formed by the addition of
two travelling waves moving in opposite directions.

A wave moving in one direction
reflects at a barrier and interferes with the incoming wave.

### Mathematical
treatment of the formation of a standing wave from two travelling waves

Consider
two travelling waves 1 and 2. Let the displacements at time t and position x be y

_{1}
and y

_{2}.

y

_{1} = a sin (

wt - kx) (say right-
left)

y

_{2} = a sin (

wt + kx) (say left-
right)

Therefore:

y_{1} + y_{2} = a sin (wt - kx) + a sin (wt + kx) = 2a sin(wt).cos(kx) = Asin (wt)
Note that this expression is composed of two
terms:

(a) sin (

wt) - this shows a varying amplitude with time
at a particular place.

(b) cos (kx) - this shows a varying amplitude with position at a
particular time.

When x = 0,

l/2 ... A is a maximum and
we have an antinode;

When x =

l/4, 3

l/4, 5

l/4 ... A is a minimum and we have a
node.

Notice that the maximum value of A is 2a.

**Student investigation**
The standing waves on a string may be studied using an experiment due to Melde and shown in Figure 2(a). A 0.5 m long rubber cord of 3 mm2 cross-section is clamped at one end, stretched to twice its length and fixed to a vibrator. The vibration generator is connected to a signal generator which can give a range of frequencies between 10 Hz and 100 Hz.

Observe the effects of slowly increasing the vibrator frequency from 10 Hz to 100 Hz and explain what you see. The experiment is best done in a darkened laboratory with the cord illuminated with a stroboscope.

Standing waves may also be investigated using the 2.8 cm wavelength microwave equipment used by most schools (Figure 2(b)).

The transmitter is set up facing a vertical metal plate about 50 cm away. If a probe detector connected to a meter is moved along the line between transmitter and plate a series of nodes and antinodes can be found. The distance between successive nodes or antinodes is half the wavelength of the microwaves.

### schoolphysics: Standing waves

To see an animation of standing waves on strings click on the animation link.